The Mathematical Work of the 2010 Fields Medalists The Notices solicited the following articles about the works of the four individuals to whom Fields Medals were awarded at the International Congress of Mathematicians in Hyderabad, India, in August 2010. The International Mathematical Union also issued news releases about the medalists’ work, and these appeared in the December 2010 Notices. —Allyn Jackson The Work of Ngô Bao Châu basis of irreducible characters bC charC h aπ trace π Thomas C. Hales = = C π In August 2010 Ngô Bao Châu was awarded a Fields Medal for his deep work relating the Hitchin fibra- for some complex coefficients bC and aπ depend- tion to the Arthur-Selberg trace formula, and in ing on h. The side of the equation with conjugacy particular for his proof of the Fundamental Lemma classes is called the geometric side of the trace for- for Lie algebras [27], [28]. mula, and the side with irreducible characters is called the spectral side. The Trace Formula When G is no longer assumed to be finite, some analysis is required. We allow G to be a A function h : G C on a finite group G is a class Lie group or, more generally, a locally compact function if h(g 1xg)→ h(x) for all x,g G. A class − topological group. The vector space V may be function is constant on= each conjugacy∈ class. A ba- infinite-dimensional so that a trace of a linear sis of the vector space of class functions is the set transformation of V need not converge. To im- of characteristic functions of conjugacy classes. prove convergence, the irreducible character is A representation of G is a homomorphism π : no longer viewed as a function but rather as a G GL(V) from G to a group of invertible linear distribution transformations→ on a complex vector space V . It follows from the matrix identity f ֏ trace π(g)f(g)dg, 1 G trace(B− AB) trace(A) = where f runs over smooth compactly supported that the function g ֏ trace(π(g)) is a class func- test functions on the group, and dg is a G-invariant tion. This function is called an irreducible charac- measure. Similarly, the characteristic function of ter if V has no proper G-stable subspace. A basic the conjugacy class is replaced with a distribution theorem in finite group theory asserts that the set that integrates a test function f over the conjugacy of irreducible characters forms a second basis of class C with respect to an invariant measure: the vector space of class functions on G. A trace formula is an equation that gives the ex- 1 (1) f ֏ f(g− xg)dg. pansion of a class function h on one side of the C equation in the basis of characteristic functions of The integral (1) is called an orbital integral. A trace conjugacy classes C and on the other side in the formula in this setting becomes an identity that ex- presses a class distribution (called an invariant dis- Thomas C. Hales is Mellon Professor of Mathematics at tribution) on the geometric side of the equation as the University of Pittsburgh. His email address is hales@ a sum of orbital integrals and on the spectral side pitt.edu. of the equation as a sum of distribution characters. March 2011 Notices of the AMS 453 The celebrated Selberg trace formula is an iden- appears in the spectral decomposition of tity of this general form for the invariant distribu- L2(G(F)Z(A) G(A)) tion associated with the representation of SL2(R) \ 2 on L (SL2(R)/ ), for a discrete subgroup . Arthur is said to be an automorphic representation. The generalized the Selberg trace formula to reductive automorphic representations (by descending to groups of higherΓ rank. Γ the quotient by G(F)) are those that encode the number-theoretic properties of the field F. The History theory of automorphic representations just for The Fundamental Lemma (FL) is a collection of the two linear groups G GL(2) and GL(1) al- ready encompasses the classical= theory of modular identities of orbital integrals that arise in connec- forms and global class field theory. tion with a trace formula. It takes several pages to There is a complex-valued function L(π,s), write all of the definitions that are needed for a s C, called an automorphic L-function, attached precise statement of the lemma [17]. Fortunately, to∈ each automorphic representation π. (The L- the significance of the lemma and the main ideas function also depends on a representation of a of the proof can be appreciated without the precise dual group, but we skip these details.) Langlands’s statement. philosophy can be summarized as two objectives: Langlands conjectured these identities in lec- tures on the trace formula in Paris in 1980 and (1) Show that many L-functions that routinely arise in number theory are automorphic. later put them in more precise form with Shelstad (2) Show that automorphic L-functions have [21], [22]. Over time, supplementary conjectures wonderful analytic properties. were formulated, including a twisted conjecture by Kottwitz and Shelstad and a weighted conjecture There are two famous examples of this philoso- by Arthur [20], [1]. Identities of orbital integrals on phy. In Riemann’s paper on the zeta function the group can be reduced to slightly easier identi- ∞ 1 ζ(s) , ties on the Lie algebra [23]. Papers by Waldspurger s = n 1 n rework the conjectures into the form eventually = used by Ngô in his solution [35], [33]. Over the he proved that it has a functional equation and years, Chaudouard, Goresky, Kottwitz, Laumon, meromorphic continuation by relating it to a θ- MacPherson, and Waldspurger, among others, series (an automorphic entity) and then using the have made fundamental contributions that led analytic properties of the θ-series. Wiles proved up to the proof of the FL or extended the results Fermat’s Last Theorem by showing that the L- afterward [24], [14], [15], [9], [10], [11]. It is hard function L(E,s) of every semistable elliptic curve to do justice to all those who have contributed over Q is automorphic. From automorphicity to a problem that has been intensively studied follows the analytic continuation and functional for decades, while giving special emphasis to the equation of L(E,s). spectacular breakthroughs by Ngô. The Arthur-Selberg trace formula has emerged With the exception of the FL for the special lin- as a general tool to reach the first objective (1) of ear group SL(n), which can be solved with rep- Langlands’s philosophy. To relate one L-function resentation theory, starting in the early 1980s all to another, two trace formulas are used in tandem plausible lines of attack on the general problem (Figure 1). An automorphic L-function can be en- have been geometric. Indeed, a geometric approach coded on the spectral side of the Arthur-Selberg is suggested by direct computations of these inte- trace formula. A second L-function is encoded on grals in special cases, which give their values as the spectral side of a second trace formula of a the number of points on hyperelliptic curves over possibly different kind, such as a topological trace formula. By equating the geometric sides of the two finite fields [19], [16]. trace formulas, identities of orbital integrals yield To motivate the FL, we must recall the bare out- identities of L-functions. The value of the FL lies in lines of the ambitious program launched by Lang- its utility. The FL can be characterized as the mini- lands in the late 1960s to use representation theory mal set of identities that must be proved in order to to understand vast tracts of number theory. Let F put the trace formula in a useable form for applica- be a finite field extension of the field of rational tions to number theory, such as those mentioned Q A numbers . The ring of adeles of F is a locally at the end of this report. compact topological ring that contains F and has the property that F embeds discretely in A with a The Hitchin Fibration compact quotient F A. The ring of adeles is a con- Ngô’s proof of the FL is based on the Hitchin fibra- venient starting point\ for the analytic treatment of tion [18]. the number field F. If G is a reductive group defined Every endomorphism A of a finite-dimensional over F with center Z, then G(F) is a discrete sub- vector space V has a characteristic polynomial group of G(A) and the quotient G(F)Z(A) G(A) \ n n 1 has finite volume. A representation π of G(A) that (2) det(t A) t a1t − an. − = + +···+ 454 Notices of the AMS Volume 58, Number 3 = geometric side 1 spectral side 1 orbital integrals || || L-functions = geometric side 2 spectral side 2 Figure 1. A pair of trace formulas can transform identities of orbital integrals into identities of L-functions. Its coefficients ai are symmetric polynomials of the dles on the spectral curve Ya. Conversely, just as eigenvalues of A. This determines a characteris- linear maps can be constructed from eigenvalues tic map χ : end(V) c, from the Lie algebra of and eigenspaces, Higgs pairs can be constructed → endomorphisms of V to the vector space c of coef- from line bundles on the spectral curve Ya. The 0 ficients (a1,...,an). This construction generalizes connected component Pic (Ya) is an abelian vari- to a characteristic map χ : g c for every reduc- ety.